Forces and Laws of Motion

Uniform Circular Motion

Uniform Circular Motion describes the motion of an object traveling at a constant speed along a circular path. Although the speed is constant, the object's velocity is continuously changing because its direction of motion is always changing. According to Newton's First Law, a change in velocity implies an acceleration. Therefore, an object in uniform circular motion is always accelerating.

Centripetal Acceleration and Force

The acceleration experienced by an object in uniform circular motion is called centripetal acceleration ($a_c$). It is always directed towards the center of the circle. Its magnitude is given by:

$a_c = \frac{v^2}{r}$

where $v$ is the constant speed of the object and $r$ is the radius of the circular path.

According to Newton's Second Law ($F_{net} = ma$), if there is an acceleration, there must be a net force causing it. The net force that causes centripetal acceleration is called the centripetal force ($F_c$). It is also directed towards the center of the circle and its magnitude is:

$F_c = ma_c = \frac{mv^2}{r}$

It's crucial to understand that 'centripetal force' is not a new fundamental force. It is the net force directed towards the center, which can be provided by familiar forces like tension, friction, gravity, or a normal force.

Newton's First Law (Law of Inertia)

Building upon the concepts of inertia, mass. Force, Sir Isaac Newton formulated three fundamental laws that form the bedrock of classical mechanics. These laws provide a comprehensive framework for analyzing the relationship between forces and motion.

Newton's First Law formalizes the concept of inertia we discussed earlier.

A body remains at rest or in uniform straight-line motion unless acted upon by a net external force.

Essentially, the First Law states that an object's velocity remains constant (which includes the case of zero velocity, i.e. being at rest) if, and only if, the net force acting on it is zero. If the net force is zero, the acceleration is zero. This reinforces that force is required to change velocity (i.e. to accelerate), not merely to maintain it. For example, a book of mass 0.5 kg resting on a table ($\vec{v} = 0$) stays at rest unless someone applies a net force to push or lift it. Similarly, a hockey puck of mass 0.1 kg sliding on near-frictionless ice ($\vec{v} = \text{constant}$) will continue sliding at that constant velocity in a straight line until a net force (like friction with rougher ice, collision with a wall, or air resistance) acts upon it.

Free Body Diagrams (FBDs)

Newton's Second Law, $\vec{F}_{net} = m\vec{a}$, is the central equation for solving dynamics problems. However, applying it correctly requires us to accurately identify all the external forces acting on the object or system of interest. A powerful and indispensable tool for achieving this is the Free Body Diagram (FBD).

An FBD is a simplified diagram representing a single object (or a system treated as a single entity) isolated from its surroundings. Its purpose is to clearly visualize all the external forces acting on that specific object. By focusing only on the object and the forces exerted on it by other objects or fields, we can systematically apply Newton's Second Law.

Constructing an accurate FBD is often the most critical step in solving a dynamics problem. Here are the general steps:

• 1. Identify the System: Clearly define the object or system of objects whose motion you want to analyze.
• 2. Isolate the Body: Draw a simple representation of the object (e.g. a dot or a box), conceptually separating it from everything else (the 'environment').
• 3. Draw External Forces: Identify every external object or field that interacts with your chosen body and exerts a force on it. Represent each force as a vector arrow originating from the object (or its center, for simplicity) and pointing in the correct direction. Common forces to consider include:
  • Gravitational Force (Weight, usually $m\vec{g}$ downwards)
  • Normal Force ($\vec{N}$, perpendicular to and away from a contact surface)
  • Tension ($\vec{T}$, along a string/rope, pulling away from the object)
  • Applied Forces ($\vec{F}_{app}$, explicitly mentioned pushes or pulls)
  • Friction ($\vec{f}$, parallel to a contact surface, opposing motion/tendency of motion)
  • Spring Force ($\vec{F}_{spring}$, opposing deformation from equilibrium)
• 4. Label Forces: Clearly label each force vector with an appropriate symbol (e.g. $W, N, T, f, F_{app}$).
• 5. Choose Coordinate Axes (Optional but Recommended): If applying Newton's Second Law in component form, draw a convenient set of coordinate axes (e.g. x-y axes). Often, aligning one axis with the direction of acceleration (if known) simplifies the calculations.

A common pitfall is including forces exerted by the object on its surroundings, or internal forces within a system. Remember, the FBD must only show external forces acting on the specific body being analyzed. For example, on an FBD for a block on a table, draw the weight (Earth on block) and the normal force (table on block).. You do not show the force exerted by the block on the table - that force belongs to the FBD of the table.

Once the FBD is correctly drawn, we can apply Newton's Second Law: $\sum \vec{F}_{ext} = m\vec{a}$. If the object is in equilibrium (at rest or moving with constant velocity), then $\vec{a} = 0$, and the vector sum of forces is zero: $\sum \vec{F}_{ext} = 0$. If the object is accelerating, the vector sum of forces equals the mass times the acceleration vector. Often, we resolve the forces and acceleration into components along the chosen coordinate axes: $\sum F_x = ma_x$ and $\sum F_y = ma_y$.

Example: Water bottle on a table

Let's illustrate with an example: Consider a water bottle of mass $m = 0.5$ kg resting in equilibrium on a horizontal table.

Introduction to Friction

Friction is a contact force that opposes the relative motion or tendency of relative motion between surfaces. It arises from the microscopic irregularities and electromagnetic interactions between the atoms of the surfaces in contact. While we often simplify problems by assuming 'smooth' surfaces (meaning frictionless), in the real world, friction is a ubiquitous and crucial force.

Types of Friction

There are two main types of friction we will consider:

• Static Friction ($f_s$): This is the force that prevents an object from starting to move. It acts in the direction opposite to the applied force that is trying to initiate motion. The magnitude of static friction is self-adjusting; it increases to match the applied force up to a certain maximum value.
• Kinetic Friction ($f_k$): This is the force that opposes the motion of an object that is already sliding. It generally has a constant magnitude for a given pair of surfaces, regardless of the speed (at least for typical speeds).

The Model of Friction

The maximum static friction (the force needed to just get an object moving) is proportional to the magnitude of the normal force $N$ between the surfaces.

$f_{s,max} = \mu_s N$

Here, $\mu_s$ is the coefficient of static friction, a dimensionless number that depends on the nature of the two surfaces in contact. The actual static friction force is always less than or equal to this maximum value: $f_s \le \mu_s N$.

The kinetic friction force is also proportional to the normal force:

$f_k = \mu_k N$

Here, $\mu_k$ is the coefficient of kinetic friction. Generally, for a given pair of surfaces, $\mu_k < \mu_s$. This is why it often takes more force to get an object moving than to keep it moving.

Introduction to Forces

Forces are the fundamental cause of all changes in motion. Understanding forces is crucial for explaining why objects move, stop, or change direction.

A force is a push or pull that can cause an object to accelerate, change direction, or deform. Forces are vector quantities, meaning they have both magnitude and direction.

Newton's Second Law

$\vec{F} = m\vec{a}$

This equation, known as Newton's Second Law, tells us that the net force acting on an object equals its mass times its acceleration.

In this chapter, we'll explore Newton's three laws of motion and learn how to analyze forces using free body diagrams.

Practice Problems

Applying Newton's Laws effectively requires practice. We start with simple examples focusing on equilibrium ($\vec{a}=0$) or straightforward acceleration, primarily involving Normal force and Tension. Use $g \approx 10 \, \text{m/s}^2$.

Practice 01: Hanging Lamp

A 2 kg lamp hangs motionless from a wire. Find the tension in the wire.

Practice 02: Two Blocks in Contact (Pushing)

Consider two blocks, with masses $m_1 = 2\,\text{kg}$ and $m_2 = 4\,\text{kg}$, placed adjacent to each other on a smooth horizontal surface. A horizontal force $F = 30\,\text{N}$ is applied to $m_1$, pushing it towards $m_2$. Find the acceleration of the system and the magnitude of the normal contact force between the blocks.

Newton's Second Law (Law of Acceleration)

While the First Law describes motion in the absence of a net force, the Second Law quantifies what happens when a net force is present.

The acceleration ($\vec{a}$) of an object is directly proportional to the net external force ($\vec{F}$) acting on it.

This fundamental law establishes the quantitative relationship between the net external force acting on an object, the object's mass, and the resulting acceleration. It states that the acceleration produced is directly proportional to the magnitude of the net force and in the same direction as the net force. Inversely proportional to the mass of the object. Mathematically, it is expressed as:

$\qquad \vec{F}_{net} = m\vec{a}$

Here, $\vec{F}_{net}$ represents the vector sum of all external forces acting on the object (also known as the resultant force), $m$ is the mass (inertia) of the object. $\vec{a}$ is the acceleration vector of the object. It is crucial to remember the vector nature of this law: the acceleration vector $\vec{a}$ always points in the same direction as the net force vector $\vec{F}_{net}$. This equation is the cornerstone of dynamics, allowing us to predict the motion of an object if we know the forces acting on it, or conversely, to determine the net force if we observe its acceleration. For example, if a net force of 10 N acts horizontally to the right on a 2 kg object initially at rest, the object will experience an acceleration of $\vec{a} = \vec{F}_{net} / m = (10 \, \text{N, right}) / (2 \, \text{kg}) = 5 \, \text{m/s}^2$ horizontally to the right.

Key Concepts

• Newton's First Law: An object at rest stays at rest and an object in motion stays in motion with the same speed and in the same direction unless acted upon by an unbalanced force. This is also known as the law of inertia.
• Newton's Second Law: The acceleration of an object as produced by a net force is directly proportional to the magnitude of the net force, in the same direction as the net force. Inversely proportional to the mass of the object. ($\vec{F}_{net} = m\vec{a}$)
• Newton's Third Law: For every action, there is an equal and opposite reaction. Forces always occur in pairs, and they act on different objects.
• Free Body Diagrams (FBDs): A crucial tool for visualizing all external forces acting on an object, allowing for the systematic application of Newton's Second Law.
• Friction: A contact force that opposes motion. Static friction prevents motion, while kinetic friction acts on moving objects.
• Uniform Circular Motion: Motion in a circle at a constant speed requires a net force, the centripetal force, directed towards the center of the circle ($F_c = mv^2/r$).

Newton's Third Law (Law of Action-Reaction)

While the first two laws focus on the motion of a single object under the influence of forces, the Third Law describes the nature of the forces themselves as interactions between objects.

For every action, there is an equal and opposite reaction. That is, if object A exerts a force on object B ($\vec{F}_{B \leftarrow A}$), then object B simultaneously exerts a force on object A ($\vec{F}_{A \leftarrow B}$) such that $\vec{F}_{A \leftarrow B} = -\vec{F}_{B \leftarrow A}$.

This law highlights several critical points about forces:

• Forces occur in pairs: Forces always arise from mutual interactions between two objects. You cannot have a single, isolated force.
• Equal in magnitude: The two forces in an action-reaction pair always have exactly the same strength.
• Opposite in direction: The two forces act along the same line but point in opposite directions.
• Act on different objects: This is crucial! The 'action' force acts on one object. The 'reaction' force acts on the other object. Because they act on different objects, they do not cancel each other out when considering the motion of a single object.
• Simultaneous: The action and reaction forces occur at the exact same time. There's no delay.

Consider these examples:

• When a person of mass 60 kg pushes horizontally against a wall with a force of 50 N (action: force by person on wall), the wall simultaneously pushes back horizontally on the person with an equal force of 50 N (reaction: force by wall on person). The person might move (if on roller skates, for instance) due to the force on them, while the wall (usually) doesn't move significantly due to the force on it (because other forces, like those from the foundation, balance it).
• When the Earth exerts a downward gravitational force on you (your weight - action: force by Earth on you), you simultaneously exert an upward gravitational force of equal magnitude on the Earth (reaction: force by you on Earth). You accelerate significantly towards the Earth because of the force on you (your mass is small), while the Earth accelerates imperceptibly towards you because of the force on it (its mass is enormous).
• When a bat hits a baseball, the bat exerts a force on the ball (action). The ball simultaneously exerts an equal and opposite force on the bat (reaction).
• When a rocket expels hot gas downwards (action: force by rocket on gas), the gas simultaneously exerts an equal and upward force on the rocket (reaction: force by gas on rocket), propelling it upwards.

In summary, a Force is an interaction between two objects or between an object and its environment, causing a change in the object's motion (acceleration) unless balanced by other forces. Based on our discussion of fundamental forces and common scenarios in mechanics, we will frequently encounter the following types of forces in this chapter:

• Gravitational Force (Weight): The downward pull exerted by a large celestial body (like Earth) on an object near its surface ($\vec{W} = m\vec{g}$).
• Normal Force ($\vec{N}$): The perpendicular contact force exerted by a surface on an object pressing against it, preventing penetration.
• Tension Force ($\vec{T}$): The pulling force transmitted through a string, rope, cable, or similar object when it is taut.
• Friction Force ($\vec{f}$): A contact force parallel to the surface that opposes relative motion or attempted relative motion between surfaces. (We will delve into this in more detail later).
• Spring Force ($\vec{F}_{spring}$): The restoring force exerted by a deformed spring, opposing the deformation. (Details to follow).